Yes they are, that is obvious.
1- It is first proven by Genetics (Haplogroup R1B-M269 for Y-DNA).
Guess who was R1B-M269? Tutankhamun, Akhenaten (his father) and Amenhotep (his grandfather).
Now 90% (up to 95%) of Basque and Irish males are R1B-M269.
As Irish is considered to be Indo-European, it leaves only 3 possibilities :
- Celtic do not belong to IE.
- Centum and Satem are two different families.
- Basque is macro-IE.
R1B is 18000 years old, which means Indic (Brahmans are R1A) separated from west europeans at the beginning of the magdalenian era.
This means proto-IE is 18000 years old, which correlated well with 445 languages (for example, Dravidian is 4500 years old and is constituted by 25 languages).
2 - It is proven by rituals : Cantabri and ancient Basque people had the same custom of couvade.
The bear cult, equally, was practised by Basques, Celts, Britons, Thraces, Daces and Getes (proto-celtic Harzh is quite identical to modern Basque Hartz(a)).
3 - It is proven by core vocabulary : for example, gibel and liver stem from TLiWETL ; emazte and woman from WEAMANTLE ; bide and way from WAITLE ; bihotz and heart from BIHEOTL ; nahi and want from WANATLI ; beste and else from HETLE ; hlore and flower from HLOWETL ; azukre and sugar from AZUKEATL, ibili and walk from AIWAILIGI ; hontz and owl from HOWTL ; oila and fowl from HOIWTLA ; ehun and hundred from EHUNTLETL ; ohe and bed from OHETL ; amets and dream from TLEAMETL ; hari and wire from HAITLI ; aizkora and axe from AITLKORAI ; peace and bake from WEATLE ; boz and voice from WOITL ; seme and son from SEOMBE ; arrasto and track from ATLASTLO, baizik and but from BAYTLIK ; adimen and mind from ADIMEINTL ; hotz and cold from HOLTL ; berro and warm from WEATLOW ; zu and you from TLU ; hemen and here from HEWENTL ; saldu and sell from SEALTL ; taula and table from TAWTLA ; bizkar and back from BAITLKATL ; atzera and rear from ATLEATLA ; lege and law from LEAWTLE ; jaungoiko and god from JAWNGOITLO ; trumoi and thunder from TLUMTLOI ; zeru and sky from ZEUTLI ; Eguberri and Yule from ETLUWETLI ; Mundua and World from WUNLTLUA ; zoru and soil from ZOITLU ; hesi and fence from HENTLI ; pareta and wall from WATLETLA ; anitz and many from HANITL ; sega and scythe from TLEITLA ; zaldi and horse from TLAWLTLI ; txakur and dog from TLAWTLUTL ; mantxango and monkey from MAWNTLAINTLO ; afer and ape from AWETL ; turtube and turtle from TLURTLUBE ; dortoka and tortoise from TLORTOITLA ; zabal and broad from TLABLAWTLA ; potolo and fat from WAWTLAWTLAW ; erle and bee from AWBEITLE ; orein and deer from TLEOTLEIN ; denbora and time from TLEINWOTLAITL ; ordu and hour from HOULTLU ; urte and year from TLEAWTLE ; Izar and star from ITLATL ; beltz and black from BEALTL ; suge and snake from TLAWTLE ; beldar and caterpillar from TLAWTLETLWITLATL ; mendi and mount/mound from MEOWNTLI ; tapoi and top from TAWPAWTLI ; pentoka and beorg from WENTLOTLA ; zuku and juice from TLUITLU ; hegi and edge from HETLI ; golko and cove from KWOTLKWO ; badia and bight from BAITLIA ; erreka and creek from KWETLEKWA ; gatz and sealt/salt from TLEATL ; irla and island from ILTLANDA ; urdin and blue from HTLULTLIN ; euri and rain from EUTLEIN ; tsitlarra and heather from TLEATLEATLA ; haize and wind from HAINTLE ; korapilo and knot from KWOTLAPITLO ; oihana and forest from HOITLANTLA ; begi and eye from WEITLI ; bisaia and face from WAITLATLA ; mingain and tongue from KWOINKWAIN ; mutur and mouth from MUTLUTL ; esan/erran and say from ETLAIN ; sudur and nose from HOTLUTLUTL ; adar and antler from ANTLEAR ; belarri and eare/ear from HEATLATLI ; hirri and city from HITLI ; oso and all from AWTLAW ; zilar and silver from TLITLABEAR ; hill and die from TLIETL ; heldu and hold from HEWLTLU ; eta and and from EANTLA ; erro and root from ETLUTL ; etxe and house from HEUTLE and so on...
A very interesting one is arrano (pronounced "arranio", meaning eagle) : arn in old english, arend in Dutch, haaran in Hittite.
Titmouse is TSIMURUNTSA in Basque, reel is HARIL, car is GARROI, bare is BILUZIK, map is MAPA ; card is KARTA ; cork is KORTXO ; up is AUPA ; easy is AISE ; end is URHENTZE ; ochre is OKRE ; carry is EKARRI ; apart is APARTE ; rock is ARROKA ; chip is TXIPA ; start is HASTE-ARTE ; March is MARTXOA ; April is APIRILA ; land is LANDA ; universe is UNIBERTSOA (verse is BERTSOA) ; mess is NAHAS-MAHAS ; tin is EZTAINU ; mare is ZAMARI ; ass is ASTO ; person is PERTSONA ; tide is ITSASALDI ; sea is ITSASOA ; minute is MINUTUA ; ruby is ERRUBI ; tuna is ATUNA ; tor is TONTOR ; machete is MATXETE ; knife is KANIBET ; ash is hautsa ; gimlet is GINBALET ; bank is BANKU ; lake is LAKUA ; ice is IZOTZ ; bean is BABARRUN ; fruit is FRUITUA ; colore is KOLOREA ("with flower") ; air is AIRE ; taste is TSASTATU ; reality is ERREALITATE ; evil is GOIBEL, the list is infinite...
But what is more important, you'll always find the same associations, for example sunflower is eguzkilore (sun-flower) ; laur is related to hlore, as four is linked to flower ; and as eight is related to light and night, zortzi is related to ortzi and sortzi ; ostegun (thursday) have the same meaning of god's day as in IE ; tomorrow is bihar (two is bi and har is cognate with mor-) ; bizitza (life) is connected to bihotz (heart), as life is correlated to liver ; as Night relates to Nil/Naught, Gaba/Gaua is linked to Gabe.
4 - The more you'll get to proto-IE level, the most you'll reach Basque.
For example, Arkwi was "white" in Tocharian ; in Basque, ARGI means "light".
UITSA or UITLA, the basque word for "water", is very close to proto-IE reconstructed form (the same for ITSASOA 'the sea').
Sometime, the sense is displaced : from example ERRIBERA means "bank of river" (BANKU is more local, ERRIBERA follows all the stream), nevertheless it is very cose to the reconstructed proto-IE form for "river".